reserve i,j,k,n for Nat;
reserve x,y,z for Tuple of n, BOOLEAN;

theorem Th22:
  i is_expressible_by n implies MUL_MOD(1,i,n) = i
proof
A1: ChangeVal_1(1,n) = 1 by Def7;
  assume i is_expressible_by n;
  then
A2: i < 2 to_power n;
  per cases;
  suppose
A3: i = 0;
    then ChangeVal_1(i,n) = 2 to_power(n) by Def7;
    then (ChangeVal_1(i,n)) mod ((2 to_power n)+1) = 2 to_power(n) by NAT_D:24
,XREAL_1:29;
    hence thesis by A1,A3,Def8;
  end;
  suppose
A4: i <> 0;
    2 to_power n < (2 to_power n)+1 by XREAL_1:29;
    then
A5: i < (2 to_power n)+1 by A2,XXREAL_0:2;
    (ChangeVal_1(i,n)) mod ((2 to_power n)+1) = i mod ((2 to_power n)+1)
    by A4,Def7;
    then (ChangeVal_1(i,n)) mod ((2 to_power n)+1) = i by A5,NAT_D:24;
    hence thesis by A2,A1,Def8;
  end;
end;
